IB Math AI HL key terms

A number written as a + bi, where a and b are real and i² = −1. The real part is a, the imaginary part is b, and arithmetic follows ordinary algebra with i² replaced by −1.

For z = a + bi the conjugate is a − bi, written z̄. Multiplying a number by its conjugate gives a real result a² + b², which is used to divide complex numbers and to find real roots of polynomials.

Writing a complex number as r(cos θ + i sin θ) or re^{iθ}, where r is the distance from the origin and θ the angle from the positive real axis. This form makes multiplication, division and powers easy.

States that [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) for integer n. It converts powers and roots of complex numbers into simple operations on modulus and argument, and underlies finding nth roots.

A sequence in which each term after the first is obtained by adding a fixed constant d, the common difference. The nth term is u₁ + (n − 1)d and the sum of n terms is n/2 times the first plus last term.

A sequence where each term is the previous term multiplied by a fixed constant r, the common ratio. The nth term is u₁rⁿ⁻¹, and sums are found with the finite formula or the infinite sum u₁/(1 − r) when |r| < 1.

Compact notation Σ from k = m to n of aₖ, meaning the sum of the terms aₖ as k runs through consecutive integers from m to n. It is used to express series and to apply standard sum formulae.

Expands (a + b)ⁿ as a sum of terms involving binomial coefficients, where the term in aⁿ⁻ʳbʳ has coefficient C(n, r). It allows any single term of a large power to be found without full expansion.

An ordered arrangement of some or all items from a set. The number of ways to arrange r items chosen from n distinct items is n!/(n − r)!, used when the order of selection matters.

A selection of items from a set where order does not matter. The number of ways to choose r items from n is C(n, r) = n!/(r!(n − r)!), also written as a binomial coefficient.

The product of all positive integers up to n, written n!, with 0! defined as 1. Factorials count arrangements of n distinct objects and appear in the formulae for permutations, combinations and the binomial theorem.

A method for proving a statement for all positive integers: show it holds for a base case, assume it holds for some general n = k, then use that assumption to prove it holds for k + 1.

Rewriting a rational expression as a sum of simpler fractions with linear or quadratic denominators. The technique simplifies integration and series expansion of rational functions with factorable denominators.

The set of all input values for which a function is defined. Restrictions come from denominators being nonzero, even roots of negatives, and logarithms of non-positive numbers; the domain must be stated when it is not all reals.

The set of all output values a function actually produces as the input runs over its domain. It is found from the graph or by analysing the function's behaviour, and changes when the domain is restricted.

The function that reverses another, written f⁻¹, so that f⁻¹(f(x)) = x. It exists only when f is one-to-one, and its graph is the reflection of f's graph in the line y = x.

A function formed by applying one function to the output of another, written (f ∘ g)(x) = f(g(x)). Its domain consists of inputs in g's domain whose outputs lie in f's domain.

A line that a curve approaches ever more closely without necessarily meeting. Vertical asymptotes arise where a denominator is zero; horizontal and oblique asymptotes describe end behaviour of rational or exponential functions.

A function that is the quotient of two polynomials. Its graph features asymptotes and possible holes, and its behaviour near excluded values and at infinity is central to sketching and analysis.

A function of the form f(x) = aˣ with a positive base a ≠ 1, or a scaled version k·aˣ. It models growth or decay at a constant proportional rate and has a horizontal asymptote.

The inverse of an exponential: logₐ x is the power to which base a must be raised to give x. Laws of logarithms turn products into sums, powers into multiples, and allow solving exponential equations.

The logarithm with base e ≈ 2.718, written ln x. It inverts the exponential function eˣ and appears throughout calculus, particularly in derivatives and integrals involving 1/x.

Changes to a graph produced by altering the function's formula: translations, reflections, stretches and combinations. Recognising how parameters shift, flip or scale a parent graph allows quick sketching without plotting points.

The quantity b² − 4ac for the quadratic ax² + bx + c. Its sign determines whether the equation has two distinct real roots, one repeated root, or no real roots, and it links to the graph's intersections with the x-axis.

A unit of angle defined so that a full turn is 2π radians. One radian subtends an arc equal in length to the radius; radians are required for arc length, sector area and calculus of trigonometric functions.

The distance along a curve between two points. For a circle of radius r and angle θ in radians, the arc length is rθ; the general formula integrates the derivative to measure curved distance.

The area of the region enclosed by two radii and an arc of a circle. With angle θ in radians it equals ½r²θ, mirroring the arc length formula and requiring radian measure.

A circle of radius 1 centred at the origin. A point at angle θ on it has coordinates (cos θ, sin θ), which defines the trigonometric functions for all angles and explains their signs and periodicity.

An equation true for all angles, such as sin²θ + cos²θ = 1 or the double-angle formulae. Identities are used to simplify expressions and to solve trigonometric equations over a given interval.

A quantity with both magnitude and direction, written as a column or as ai + bj + ck. Vectors support addition, scalar multiplication, and describe displacement, velocity and lines in space.

For vectors a and b, the product |a||b|cos θ, computed from components as the sum of corresponding products. It finds the angle between vectors, tests perpendicularity, and projects one vector onto another.

Writing a line as r = a + tb, where a is the position vector of a point on the line, b a direction vector and t a scalar parameter. It extends naturally to lines in three dimensions.

A measure of spread equal to the square root of the variance, expressed in the same units as the data. It quantifies typical distance of values from the mean and is affected by outliers.

The average of squared deviations from the mean, equal to the standard deviation squared. It is additive for independent variables and appears in the formulae for combining and scaling random variables.

The long-run mean of a random variable, found by summing each value times its probability, or integrating x times the density. It is denoted E(X) and is linear: E(aX + b) = aE(X) + b.

The distribution of the number of successes in n independent trials, each with the same probability p of success. Probabilities use C(n, r)pʳ(1 − p)ⁿ⁻ʳ, with mean np and variance np(1 − p).

A continuous, symmetric, bell-shaped distribution defined by its mean μ and standard deviation σ. Probabilities are found from the standard normal curve, often after converting to z-scores.

The number of standard deviations a value lies from the mean: z = (x − μ)/σ. Standardising allows comparison across different distributions and lookup of probabilities on the standard normal curve.

The probability of event A given that B has occurred, written P(A|B) = P(A ∩ B)/P(B) when P(B) > 0. It underlies tree diagrams and tests for independence.

Two events where the occurrence of one does not change the probability of the other, so P(A ∩ B) = P(A)P(B). This is distinct from mutually exclusive events, which cannot occur together.

A rule for reversing conditional probability: P(A|B) = P(B|A)P(A)/P(B). It updates a prior probability with new evidence and is often applied through tree diagrams or two-way tables.

The instantaneous rate of change of a function, defined as the limit of a difference quotient and written f′(x) or dy/dx. It gives the gradient of the tangent and is the basis of optimisation and curve sketching.

A rule for differentiating composite functions: the derivative of f(g(x)) is f′(g(x))·g′(x). It is essential for functions such as (2x + 1)⁵, e^{3x} and ln(cos x).

A rule for differentiating a product: if y = uv then dy/dx = u′v + uv′. It applies where two functions of x are multiplied and cannot be simplified by expansion.

A rule for differentiating a fraction: if y = u/v then dy/dx = (u′v − uv′)/v². It handles ratios of functions such as polynomials over exponentials or trigonometric expressions.

A point where the derivative equals zero, giving a horizontal tangent. Its nature - local maximum, local minimum or point of inflection - is determined by the sign change of the derivative or the second derivative test.

Using calculus to find the greatest or least value of a quantity, often subject to a constraint. It involves forming a function of one variable, differentiating, solving for stationary points and justifying the optimum.

The limit of a sum of areas under a curve between two limits, written with lower and upper bounds. It gives signed area and is evaluated using an antiderivative, with the fundamental theorem of calculus linking it to differentiation.

A function whose derivative is the given function, also called an indefinite integral and written with + C. Finding antiderivatives reverses differentiation and is the key step in evaluating definite integrals.

The region between a graph and the x-axis, computed by a definite integral. Regions below the axis count as negative, so separate integrals or absolute values are needed when the curve crosses the axis.